The adjoint of the matrix A = $\left|\begin{array}{ccc}1& 0& 2\\ 2& 1& 0\\ 0& 3& 1\end{array}\right| \mathrm{is} :$

• $\left[\begin{array}{ccc} - 1& 6& 2\\ - 2& 1& - 4\\ 6& 3& 1\end{array}\right]$

• $\left[\begin{array}{ccc}1& 6& - 2\\ - 2& 1& 4\\ 6& - 3& 1\end{array}\right]$

• $\left[\begin{array}{ccc} 6& 1& 2\\ 4& - 1& 2\\ 6& 3& - 1\end{array}\right]$

• $\left[\begin{array}{ccc} - 6& 2& 1\\ 4& - 2& 1\\ 3& 1& - 6\end{array}\right]$

If $\mathrm{p} + \mathrm{q} + \mathrm{r} = \mathrm{a} + \mathrm{b} + \mathrm{c} = 0$, then the determinant

$\left|\begin{array}{ccc}\mathrm{pa}& \mathrm{qb}& \mathrm{rc}\\ \mathrm{qc}& \mathrm{ra}& \mathrm{pb}\\ \mathrm{rb}& \mathrm{pc}& \mathrm{qa}\end{array}\right| = ?$

• 0

• 1

• pa + qb + rc

• pa + qb + rc + a + b + c

Which of the following determinants have value 'zero' ?

$$\begin{gathered} \mathrm{I}. \left|\begin{array}{ccc}41& 1& 5\\ 79& 7& 9\\ 29& 5& 3\end{array}\right| \\ \mathrm{II}. \left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b} + \mathrm{c}\\ 1& \mathrm{b}& \mathrm{c} + \mathrm{a}\\ 1& \mathrm{c}& \mathrm{a} + \mathrm{b}\end{array}\right| \\ \mathrm{III}. \left|\begin{array}{ccc}0& \mathrm{c}& \mathrm{b}\\ - \mathrm{c}& 0& \mathrm{a}\\ - \mathrm{b}& - \mathrm{a}& 0\end{array}\right| \end{gathered}$$

Select the correct answer using the code given below

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

$\mathrm{Let} {\mathrm{ax}}^3 +\hspace{0.17em}{\mathrm{bx}}^2 +\mathrm{cx} + \mathrm{d} = \left|\begin{array}{ccc}\mathrm{x} + 1& 2\mathrm{x}& 3\mathrm{x}\\ 2\mathrm{x} + 3& \mathrm{x} + 1& \mathrm{x}\\ 2 - \mathrm{x}& 3\mathrm{x} + 4& 5\mathrm{x} - 1\end{array}\right|, \mathrm{then}$

What is the value of c ?

•  - 1

• 34

• 35

• 50

$\mathrm{Let} {\mathrm{ax}}^3 +\hspace{0.17em}{\mathrm{bx}}^2 +\mathrm{cx} + \mathrm{d} = \left|\begin{array}{ccc}\mathrm{x} + 1& 2\mathrm{x}& 3\mathrm{x}\\ 2\mathrm{x} + 3& \mathrm{x} + 1& \mathrm{x}\\ 2 - \mathrm{x}& 3\mathrm{x} + 4& 5\mathrm{x} - 1\end{array}\right|, \mathrm{then}$

What is the value of a + b + c + d ?

• 62

• 63

• 65

• 68

26.

If $\mathrm{m} = \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \mathrm{and} \mathrm{n} = \left[\begin{array}{cc}0& 1\\ - 1& 0\end{array}\right],$then what is the value of the determinant of $\mathrm{mcos}\left(\mathrm{\theta }\right) - \mathrm{nsin}\left(\mathrm{\theta }\right) ?$

•  - 1

• 0

• 1

• 2

If x + a + b + c = 0, then what is the value of $\left|\begin{array}{ccc}\mathrm{x} + \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{a}& \mathrm{x} + \mathrm{b}& \mathrm{c}\\ \mathrm{a}& \mathrm{b}& \mathrm{x} +\mathrm{c}\end{array}\right|$ = ?

• 0

• (a + b + c)²

• a² + b² + c²

• a + b + c - 2

What are the values of x that satisfy the equation 

$\left|\begin{array}{ccc}\mathrm{x}& 0& 2\\ 2\mathrm{x}& 2& 1\\ 1& 1& 1\end{array}\right| + \left|\begin{array}{ccc}3\mathrm{x}& 0& 2\\ {\mathrm{x}}^2& 2& 1\\ 0& 1& 1\end{array}\right| = 0$

• $- 2 \pm \sqrt{3}$

• $- 1 \pm \sqrt{3}$

• $- 1 \pm \sqrt{6}$

• $- 2 \pm \sqrt{6}$

What is the value of the determinant $\left|\begin{array}{ccc}1!& 2!& 3!\\ 2!& 3!& 4!\\ 3!& 4!& 5!\end{array}\right|$ ?

• 0

• 12

• 24

• 36

What is the value of the determinant $\left|\begin{array}{ccc}\mathrm{i}& {\mathrm{i}}^2& {\mathrm{i}}^3\\ {\mathrm{i}}^4& {\mathrm{i}}^6& {\mathrm{i}}^8\\ {\mathrm{i}}^9& {\mathrm{i}}^{12}& {\mathrm{i}}^{15}\end{array}\right|$

• 0

•  - 2

• 4i

• - 4i